The graph of the line $x+y=b$ is a perpendicular bisector of the line segment from $(0,3)$ to $(6,9)$. What is the value of b?
Solution: If the line $x+y=b$ is the perpendicular bisector of the segment from $(0,3)$ to $(6,9)$, it must pass through the midpoint of this segment.  The midpoint is: $$\left(\frac{0+6}{2},\frac{3+9}{2}\right)=(3,6)$$This point lies on the line $x+y=b$, so we must have $3+6=b\Rightarrow b=\boxed{9}$.